Effect of Dean Number on Heat Transfer Coefficients in Coiled Tube Agitated Vessel

EFFECT OF DEAN NUMBER ON HEAT TRANSFER COEFFICIENTS IN COILED TUBE AGITATED VESSEL

Ansar Ali sk , Vipin Sharma Meerut Institute of Engineering and Technology, Meerut, India

Ashok Kumar Ojha Saraswati Institute of Engineering and Technology, Hapur, INDIA

ABSTRACT -The purpose of the experimental phase of the present investigation is to obtain data for heat transfer to purely viscous non-Newtonian fluids in coil and agitated vessel. The set up consisted of a flat bottomed cylindrical test vessel ‘A’ of 45.25 cm inner diameter and 60 cm height, made from 1/8 inch thick copper sheet. An annular space ‘F’ around the test vessel was provided by surrounding it with another cylinder of 44 cm height and made from 1/8 inch thick G.I. sheet. The jacket and the vessel assembly was adequately insulated by keeping it in a wooden tank and then filling the surrounding space with glass wool and thermocouple. Three aqueous solutions containing 0.5% (n=0.973), 1% (n= 0.851) and 2% (n=0.793) CMC and water were studied in the coil and 1%, 2% and 4% (n=0.698) aqueous solutions of CMC in the vessel. In our present study hic / his Vs ND2 , NNuic / NNuis Vs ND2 were studied.

KEYWORDS -Dean Number, Agitated Vessel, helical coil, CMC solution, heat transfer coefficient

INTRODUCTION Several investigators have studied the average heat transfer from helical coils in the mixing vessels. The heat transfer rate at the coil surface may be written as 휕푇 푄 = −푘 ∫ ( 푖) 푑퐴 (1.1) 퐴 휕푥 푠 Where A is the total surface area and x is the distance measured outward from and normal to the coil surface. Flow from the impeller passing around the coil surface, forms boundary layer which in turn offers resistance to heat transfer. Thus the velocity and temperature profile will depend on the impeller Reynolds number, Prandtl number and the diameter of the coil tube. On the ′′ ′′ basis of the discussions made it can be shown that the temperature gradient is a function of (푁푅푒푎) and(푁푃푟푎). Thus, in order to 퐷푡 describe the heat transfer from the coil surface, a correlation can be developed between the Nusselt number(ℎ표푐 ⁄푘), Reynolds ′′ ′′ number, (푁푅푒푎), and Prandtl number, (푁푃푟푎). For coils in mixing vessel and for given geometrical situation and isothermal condition this results in the following equation ℎ 퐷 표푐 표 = 퐶 (푁′′ )푏3 (푁′′ )푏4 (1.2) 푘 2 푅푒푎 푃푟푎 Laminar forced convection in curved pipes has received considerable attention in recent years. From the work of Akiyama and his co-workers (1971), Dravid (1971), Kubair and Kuloor (1966) and Rajshekhran et al. (1966), it is seen that very little work has been done for non-Newtonian fluids in curved pipes. The existing heat transfer results in the literature for fully developed laminar forced convection in curved pipe with uniform wall temperature are rather limited and incomplete. For Newtonian fluids, perturbation method was applied by Maekawa for extremely low Dean Number. The boundary layer approximation near the wall was presented by Mori and Nakayama (1965) for high Dean Number of order one. Dravid et al. (1971) presented the numerical results in the thermal entrance region for Dean Number less than 225 and Npr = 5. The recent and improved results of Akiyama (1971) shows that the ratio of heat transfer coefficients in coil and in straight pipe is a function of Dean Number as well as Prandtl number. Thus there is possibility of obtaining a suitable correlation of the following form 푁푁푢푖푐 − 1 = ∅1 (푁퐷 , 푁푃푟) (1.3) 푁푁푢푖푠 The advantage of the above form of correlation is that for small ND, the equation satisfies the condition that NNuic ≈ NNuis for very very small ND. NNuis may be calculated for isothermal laminar heat transfer in straight tube for the case of uniform wall temperature and parabolic velocity distribution by the following correlation: 1 푁푁푢푖푠 = 1.75 (푁퐺푧)3 (1.4) For non-Newtonian fluids equations (1.3) and (1.4) take the following form: 3푛+1 1 푁 = 1.75 ( ) (푁 )3 (1.5) 푁푢푖푠 4푛 퐺푍 Here again it is important to note that Dean Number contains Reynolds number and the viscosity term appears in both the Reynolds and Prandtl numbers. It is suggested that the effective viscosity µ2 at the shear stress prevailing at the wall should be used for evaluating the Reynolds and Prandtl numbers. From the above discussions it may be concluded that the correlation may be written as below:

JETIRA006338 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 108

푁푁푢푖푐 푏5 푏6 − 1 = 퐶3 푁퐷 푁푃푟 (1.6) 푁푁푢푖푠 2 2

EXPERIMENT The following observations were recorded. Inlet and exit fluid temperatures in the jacket and the coil, amount of the water from the jacket and the coil collected separately for a known time interval, rpm of the agitator, temperature of the fluid in the agitated vessel and that in the storage tanks. The readings were duplicated to ensure the steady state and to eliminate any error in the measurement. Similar measurements were made by varying the flow rate in the coil and then by varying the rotational speed of the agitator. The above procedure was repeated with all the non-Newtonian fluids for different agitational speeds and coil flow rates. Adequate care was taken to maintain constant temperatures in both the storage tanks during a particular run period. The following sets of the experiments were conducted: 1. In order to check the accuracy of the experimental set up, data were taken on Newtonian fluids for different flow rates and the agitational speeds. 2. In order to propose the laminar and turbulent heat transfer correlations for the flow of power law non-Newtonian fluids in coils, the data were taken with four aqueous CMC solutions at different flow rates. 3. Data were taken with different agitational speeds to propose correlations for jacket and coil outside heat transfer. 4. Pressure drops were measured during each coil flow rate studied, to present a correlation for momentum transfer to non- Newtonian fluids flowing through the coil.

RESULT AND DISCUSSION

The ratio of friction factor, fc, for coil to that of straight tube, fSL, evaluated at Reynolds number,푁푅푒2 , from equation푓푆퐿 =

16⁄푁푅푒2 is plotted against 푁퐷2 in figure (1). Three non-Newtonian fluids: 0.5 (n= 0.937), 1(n= 0.851) and 2% (n= 0.793) CMC-A in water, representatives of pseudoplastic fluids obeying power law, and water, a Newtonian fluid were investigated. Most of the data obtained were in laminar region and very few data of non-Newtonian fluids could fall in turbulent region. Due to lack of turbulent flow data for non-Newtonian fluids in coil, only laminar region data are processed and correlated. 100

SL2 10 symbol fluid 4.0

c (fc/fsL2) -1=0.033[logND2] f 0.5% CMC A 0.937 1% CMC A 0.851 2% CMC A 0.793

1 10 100 1000 10000

ND2

FIG 1: laminar flow friction factor in helical coils as a function of dean number for non-Newtonian fluids In order to find the effect ND2 on the ratio of heat transfer in coil to heat transfer in straight pipe, the first attempt was to plot 푁푁푢푖푐 the ratio ( ) against ND2 on a logarithmic scale as shown in figure (2). 푁푁푢푖푠 100

symbol fluid n

10 0.5% CMC A 0.937 1% CMC A 0.851

NUIC NUIS N N 2% CMC A 0.793

1 10 100 ND2 1000 10000

FIG 2: Variation of nnuic/ nnuis with nd2 for heat transfer to non-Newtonian fluids

JETIRA006338 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 109

No direct relationship is seen between the two variables in this manner. At very low Dean Number or very small curvature, 푁푁푢푖푐 effect of secondary flow will be negligible and then NNuic will be equal to NNuis. Therefore, the next attempt was to plot ( ) − 푁푁푢푖푠 1 against ND2. This plot is shown in figure (3). 10

]

symbol fluid n 1 0.5% CMC A 0.937

NUIS 1% CMC A 0.851

/N 2% CMC A 0.793

NUIC 0.1

[N 10 100 1000 10000

ND2

FIG 3: Variation of [(nnuic/nnuis ) -1] with nd2 for heat transfer to non-Newtonian fluids

No net conclusion could be drawn even from this figure except for some qualitative ones. The heat transfer coefficient in coil 푁푁푢푖푐 is always higher than in straight pipe as it is seen that is always greater than unity. The ratio hic/ his increases with Dean 푁푁푢푖푠 number, ND2 . These data do not follow any particular trend. This may be explained by the fact that due to non Newtonian behavior Prandtl number changes by change of flow rate affecting the heat transfer rate. Data plotted for three fluids in figure 3.3. have wide range of Prandtl number. However, an exponent of Dean Number ½ correlated the data much better. This trial has been made on the basis of theoretical results presented by previous investigators.

CONCLUSION As a result of the present investigation on the heat transfer to agitated fluids flowing through immersed helical coils in jacketed vessels, the following conclusions are drawn: 1. For small temperature driving forces the non isothermal correlation is negligible. 2. The heat transfer data for agitated Newtonian and non-Newtonian fluids have been successfully correlated by using the viscosity of the fluid evaluated at the impeller tip assuming a cylinder of diameter equal to that of impeller rotating in an infinite fluid. Data of 1, 2 and 4% CMC-A, for three impeller diameters, have been correlated by the following equation: For coil 0.1 2/3 1/3 퐷푎 푁푁푢표푐 = 0.036 푁′′푅푒푎 푁′′푃푟푎 ( ) 퐷푐

(Standard deviation 15.03%) Using the above concepts of Reynolds and Prandtl numbers it is also possible to correlate the available published data for other non-Newtonian fluids obtained with different impeller geometries. 3. The use of same Reynolds and Prandtl number expressions in correlation of jacket side and coil side heat transfer data provides a method of comparison between hj and hoc. The ratio hj/hoc is given by ℎ 0.1 푗 퐷푐 퐷푡 = 8.39 ( ⁄퐷 ) ( ⁄퐷 ) ℎ표푐 푇 푇

4. Under the identical flow conditions, laminar flow heat transfer rate in helical coils are higher than those in straight pipes. 푁푁푢푖푐 The ratio ⁄ is found to be a function of 푁퐷 and 푁푝푟 . The data obtained have been correlated by the following 푁푁푢푖푠 2 2 equation. 푁푁푢푖푐 1/2 0.12 ⁄ = 1 + 0.0666 푁 푁푝푟 푁푁푢푖푠 퐷2 2 (Standard deviation 15.52%)

24 < 푁퐷2 < 2000, 40 < 푁푝푟2<225 and 0.793

NOMENCLATURE b3,b4 constants in equation (1.2) b5,b6 constants in equation (1.6) Da agitator diameter, cm Dc diameter of the coil helix,cm

JETIRA006338 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 110

Dt inner diameter of the straight or coil tube Do outer diameter of the straight or coil tube DT diameter of the agitated vessel fc friction factot in coil fsL laminar flow friction factor in straight pipe 2 0 hj heat transfer coefficient for jacketed vessel wall to fluid, Kcal/hr m C 2 0 hoc coil outside heat transfer coefficient , kcal/hr m c 2 0 hic coil inside heat transfer coefficient , kcal/hr m c k thermal conductivity n flow behavior index n’ generalized flow behavior index r radial distance, cm t time, sec Tci inlet fluid temperature in the coil, 0C Tco outlet fluid temperature in the jacket, 0C Ti inlet temperature Ts surface temperatue U average velocity,cm 2 Uj jacket overall heat transfer coefficient, Kcal/hr m 2 0 UOC coil overall heat transfer coefficient, Kcal/hrm c u local velocity in the x direction at r or y, cm/sec

Dimensionless groups NNu Nusselt number, h D/k NNuj Nusselt number, hj DT/k NNuoc Nusselt number, hoc Do/k NNuic Nusselt number, hic Dt/k NNUIS Nusselt number, his Dt/k NRe Reynolds number, DtUρ/µ NRe2 Reynolds number defined by equation NRe” Reynolds number defined by equation N”Rea Reynolds number defined by equation NPra Prandtl number defined by equation Npr Prandtl number N’’pr Prandtl number defined by equation ND Dean number Nprd” prandtl number defined by equation

Subscript j jacket c coil I inner, inlet m mean value o outlet, outer y yield SL laminar flow in straight pipe

JETIRA006338 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 111

REFERENCES (1) Nakayama, W. & Mori, Y., Int. j.Heat Mass Trans., 8, 67 (1965). (2) Kubair,V. & Kuloor,N.R., “flow of Newtonian fluids in archimedian spiral tube coils: correlation of the laminar, transition and turbulent flows”Ind.J1.Tech., 4, 3 (1966). (3) Akiyama,M. & Cheng,K.C., “boundary vorticity method for laminar forced convected heat transfer in curved pipe” Int.J. Heat Mass Trans., 14,Pt.2, 1659 (1971). (4) Dravid, A.N., Smith, K.A., Merrill, E.W. & Brian, P.L.T., “effect of secondary fluids on laminar flow heat transfer in helical coiled tubes” A.I.Ch.E.J., 17, 1114 (1971). (5) Liou TM: Flow visualization and LDV measurement of fully developed laminar flow in helically coiled tubes. Exp Fluids , 12, 332-338 (1992). (6) Enio Kumpinskyt, “Heat-Transfer Coefficients in Agitated Vessels. Sensible Heat Models”, Ind. Eng. Chem. Res. 1995,34, 4571-4576 (7) Mujawar,B.A., & Raja Rao,M., “flow of non Newtinian fluids through helical coils”, industrial and engineering chemistry process design and development vol.17, 22(1997). (8) Rai C. L. , Devotta I. , Rao P. G. ; Heat transfer to viscous Newtonian and non-Newtonian fluids using helical ribbon agitator, Chemical engineering journal , vol. 79, no1, pp. 73-77 ISSN 1385-8947(2000) (9) Zalc, J.M., Szalai, E.S., Alvarez, M.M., Muzzio, F.J., Using CFD to understand chaotic mixing in laminar stirred tanks, A.I.Ch.E. Journal, 48(10), 2124-2134(2002) (10) Yoon, H.S., Balachandar, S., Ha, M.Y., Kar. K.,Large eddy simulation of flow in a stirred tank, Transactions of the A.S.M.E., 125, 486-499 (2003). (11) Yeoh, S.L., Papadakis, G., Yianneskis, M., Determination of mixing time and degree of homogeneity in stirred vessels with large eddy simulation, Chemical Engineering Science, 60, 2293-2302(2005) (12) Kaya O, Teke I: Turbulent forced convection in helically coiled square duct with one uniform temperature and three adiabatic walls. Heat Mass Transf , 42, 129-137(2005). (13) S.S. Pawar , Vivek K. Sunnapwar “Experimental studies on heat transfer to Newtonian and non-Newtonian fluids in helical coils with laminar and turbulent flow” Experimental Thermal and Fluid Science 44 (2013 ) 792–804 (14) Mohammad Sharif ASL, Davood Toghraie , Ahamad Reza Azimiam ” Numerical Investigation on Heat Transfer Coefficient Enhancement of Non-Newtonian Nanofluid in the Turbulent Flow Inside A Tube” Indian J.Sci.Res.1(2) : 363- 369, 2014 , ISSN : 0976-2876 (15) N.Fedal CastroB.ChitraR.Pushpalatha, S.Sudalai,” Heat transfer Effects for two different impellers using Newtonian and Non-Newtonian fluids in an Agitated Vessel” International Journal of ChemTech Research CODEN (USA): IJCRGG ISSN: 0974-4290 Vol.7, No.6, pp 2802-2808, 2014-2015.

JETIRA006338 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 112